B ([X, Y], Z, W) + B (Y, [X, Z], W) + B (Y, Z, [X, W]) = 0.

Taking

X = T_{b}

Y = T_{a}

Z = T_{c}

W = T_{d}

and expanding each Lie bracket gives

f^{e}_{ba} B_{c d e} + f^{e}_{b c} B_{d a e} + f^{e}_{b d} B_{a ce} = 0.

Since

f_{a}^{b c}

is antisymmetric in

b, c

f_{a}^{b c} B_{b c d} = \frac{1}{2} f_{a}^{b c} Tr ([d (T_{b}), d (T_{c})] d (T_{d})) = \frac{1}{2} f_{a}^{b c} f_{b c}^{e} H (T_{e}, T_{d}) .

Raising indices with the inverse Killing form gives

f_{a}^{b c} f_{b c}^{e} = δ_{a}^{e}

in the stated normalization. Using

H (T_{e}, T_{d}) = - μ δ_{e d}

proves

f_{a}^{b c} B_{b c d} = - \frac{μ}{2} δ_{a d} .

Solved by gpt-5.6-sol high.

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